ELASTICITY-AIRYS STRESS FUNCTIONBYNaziya I. Ghanchi.Roll No.-03M. E.- StructuresSaraswati college Of Engineering.

30th august 2013.*

Introduction Displacement FormulationEliminate the stresses and strains from the general system of equations. This generates a system of three equations for the three unknown displacement components. Stress FormulationEliminate the displacements and strains from the general system of equations. This generates a system of six equations and for the six unknown stress components. The General System of Elasticity Field Equations of 15 Equations for 15 Unknowns Is Very Difficult to Solve for Most Meaningful Problems, and So Modified Formulations Have Been Developed.

Two and Three Dimensional ProblemsxyzxyzThree-DimensionalTwo-DimensionalxyzSpherical Cavity

Solution to Elasticity ProblemsIn principle all practical problems 3D problems very complex difficult to handle. Reasonable assumptions bring down the complexity in the problem reduces a 3dimension => 2D problem

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Solution to Elasticity ProblemsEven Using Displacement and Stress Formulations Three-Dimensional Problems Are Difficult to Solve!So Most Solutions Are Developed for Two-Dimensional Problems

The Stress Function MethodAn effective way of dealing with many two dimensional problems is to introduce a new unknown, the Airys stress function () , an idea brought to us by George Airy in 1862.

*George Biddell Airy (1801-1892)

Airys stress function

In stress based formulation one equation in terms of normal stresses and two equilibrium equations for finding out the complete stress state are require.To Reduce the complexity of equations introduce a new filed variable call Airys stress function.Aim reduce the governing equations from three to one.Airys stress function (ASF) defined for stress based formulations both plane stress and plane strain.*

Airy stress function methodPlane problems with no body forcesStress FormulationAiry RepresentationBiharmonic Governing Equation(Single Equation with Single Unknown)

Solutions to Plane ProblemsPolar CoordinatesTraction Boundary ConditionsAiry RepresentationBiharmonic Governing Equation

Solutions to Plane ProblemsCartesian CoordinatesAiry RepresentationBiharmonic Governing EquationTraction Boundary Conditions

Hollow Cylinder Under Uniform Boundary PressureBoundary Conditions General AxisymmetricStress SolutionUsing Strain Displacement Relations and Hookes Law for plane strain gives the radial displacement

Advantages of Airys Stress functionsIt is a very popular method for the solution of the plane problem. It reduce the general formulation to a single equation in terms of a single unknown. The general idea is to develop a stress field that satisfies equilibrium and yields a single governing equation from the compatibility equations. The obtained equilibrium equation can be solved analytically in closed-form.*

*Limitations of Airys Stress functions

The Airys Stress function is applicable only to plane strain or plane stress problem. The Airys Stress function approach works best for problems where a solid is subjected to prescribed tractions on its boundary, rather than prescribed displacements.

* References

[1] Pie Chi Chou and Nicholas J.Pagano, Elasticity Tensor, dyadic, and engineering Approaches, New York, Dover publications, Inc., 1992.

[2] Timoshenko and Goodier, Theory of Elasticity, New York, McGraw-Hill, 1970 [3] Adel S. Saada, Elasticity Theory and applications, Florida, Krieger Publication Company, 1993.

[4] George E. Mase, Theory and problems of Continuum Mechanics, New york,Schaums outline series of McGraw- Hill, 1970.

[5] Daniel Frederick and Tien Sun Chang, Continuum Mechanics, Bostan, Allyn And Beacon, Inc. 1965.

[6] www.engin.brown.edu/courses/en175/notes/airy/airy.htm